Higher-spin masslessSmatrices in four dimensions

Abstract

On-shell, analytic $S$-matrix elements in massless theories are constructed from a finite set of primitive three-point amplitudes, which are fixed by Poincar\'e invariance up to an overall numerical constant. We classify all such three-point amplitudes in four dimensions. Imposing the simplest incarnation of locality and unitarity on four-particle amplitudes constructed from these three-particle amplitudes rules out all but an extremely small subset of interactions among higher-spin massless states. Notably, the equivalence principle and the Weinberg-Witten theorem are simple corollaries of this principle. Further, no massless states with helicity larger than two may consistently interact with massless gravitons. Chromodynamics, electrodynamics, Yukawa and ${\ensuremath{\phi}}^{3}$ theories are the only marginal and relevant interactions between massless states. Finally, we show that supersymmetry naturally emerges as a consistency condition on four-particle amplitudes involving spin-$3/2$ states, which must always interact gravitationally.